Solve each equation. See Example 7. x^3/2 = 125

Verified step by step guidance

1. Identify the equation given: \(x^{\frac{3}{2}} = 125\).
2. Recall that \(x^{\frac{3}{2}}\) means \((x^{\frac{1}{2}})^3\) or \((\sqrt{x})^3\).
3. To isolate \(x\), raise both sides of the equation to the reciprocal power of \(\frac{3}{2}\), which is \(\frac{2}{3}\), so apply \(\left( \cdot \right)^{\frac{2}{3}}\) to both sides: \(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 125^{\frac{2}{3}}\).
4. Simplify the left side using the property of exponents: \(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = x^{\left(\frac{3}{2} \times \frac{2}{3}\right)} = x^1 = x\).
5. Evaluate the right side \$125^{\frac{2}{3}}$ by first finding the cube root of 125, then squaring the result.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, x^(3/2) means the square root of x raised to the third power, or equivalently, (√x)^3. Understanding how to manipulate these exponents is essential for solving equations involving fractional powers.

Isolating the Variable

To solve equations, isolate the variable term by performing inverse operations. In this case, you may need to raise both sides of the equation to the reciprocal power to eliminate the rational exponent and solve for x.

Properties of Exponents

Properties such as (a^m)^n = a^(mn) and the ability to raise both sides of an equation to a power are crucial. These rules allow you to simplify expressions and solve equations involving exponents systematically.